You seem to use $n$ regarding the Schönhage–Strassen algorithm to denote the bit-length (number of digits up to a constant factor). But the Question posted (some time ago) uses $n$ to denote the argument of the factorial function. I don't think your Answer holds together as a (late) response.

This Javascript applet by Prof. Lauren Williams may give you a chance to refine what you are asking for.

In the past I've done this by recursive descent on the largest part of $n$. Does more need to be said about this?

Your test case seems to be small enough to show Readers using matrix notation (MathJax).

Note that one of the updates, deleting a row of $A$, affects the product $AB$ in an especially simple way: delete the corresponding row of the product. (This changes the shape of the matrix $A$ and the product $AB$, which might not be what the OP means.) Similarly for adding a new row (insert new product row in the respecting place).

Your title mentions regular polygon. But the example of a right triangle and the closing sentence concerning arbitrary polygon do not confine discussion to regular polygons.

@zwol: Perhaps you should explain what ability you do have to "feed $F$" points. Typically when a function is described algorithmically as "a black box", it means precisely that you have the ability to give $F$ inputs and observe the outputs. There is a basic result as to a homeomorphism known as invariance of domain. To wit: if $mn \neq m' n'$, no homeomorphism is possible.

A raster image matrix does not in itself provide a definition of adjacency. There are adjacency matrices defined for directed or undirected graphs, but your post does not seem concerned with graph theory. Please clarify what things are adjacent

Do you assume the function $f(\theta,\phi)$ is periodic for $\phi \in [0,2\pi]$?

I hope for the OP to supply more details about the precision required, in which case we could flesh out your idea of an ideal implementation of $\arctan(\sqrt x)$.

Indeed the summing of the $n$ components to get $x$ for $\arctan(\sqrt x)$ could be a dominating aspect for large enough $n$. The summation might be susceptible to optimization by "updating". There's not enough detail in the Question's body to discuss that idea. More notation would be welcome (use $\LaTeX$ for typeset mathematics).

@nicoguaro See Tolman-Oppenheimer-Volkoff equation of state which models the core of a neutron star.

The first thing to clarify is what purpose the computed inverses will serve. Most applications are better processed by solving linear systems rather than computing an inverse matrix.

I've written up an answer, which includes a small example of $A,B$ for which there is no generalized eigenvalue solution. Read it over and comment if you still have questions.

@nicoguaro: I'm happy to do so, and hoped that the OP would first clarify with any context that promises solutions exist. Real world problems are often so.

There may be no solutions as demonstrated by taking $A,B$ to be (say) $2\times 1$ columns that are linearly independent. Candidates for $\lambda$ are identifiable from $A^* A x = \lambda A^* B x$ with square matrices.